Chapter 14 Exercises: Advanced Bankroll and Staking Strategies

Instructions: Complete all exercises in the parts assigned by your instructor. Show all work for derivation and calculation problems. For programming challenges, include comments explaining your logic and provide sample output. Where simulations are required, use at least 10,000 iterations unless otherwise specified.

Part A: Kelly Criterion Theory and Extensions (10 exercises, 5 points each)

Exercise A.1 --- Full Kelly Derivation for Even Money

Starting from the growth rate function $G(f) = p \log(1 + f) + q \log(1 - f)$ for an even-money bet ($b = 1$):

(a) Derive the optimal Kelly fraction $f^*$ by taking the derivative and setting it to zero.

(b) Verify that the second derivative is negative at $f^*$, confirming it is a maximum.

(c) Calculate $f^*$ and the corresponding growth rate $G(f^*)$ for $p = 0.55$.

(d) Plot (or describe) the growth rate $G(f)$ as a function of $f$ for $f \in [0, 1]$ when $p = 0.55$. At what fraction $f$ does the growth rate become negative?

Exercise A.2 --- Kelly for Three-Outcome Bets

Consider a bet with three outcomes: win $b_1$ with probability $p_1$, win $b_2$ with probability $p_2$, and lose the stake with probability $q = 1 - p_1 - p_2$.

(a) Write the growth rate function $G(f)$ for this three-outcome bet.

(b) Derive the first-order condition for the optimal fraction $f^*$.

(c) Apply your formula to a soccer match draw-no-bet where: win $+1.0$ with probability 0.45, push (return stake) with probability 0.25, lose with probability 0.30. What is the optimal Kelly fraction?

(d) Compare this to the standard two-outcome Kelly formula with $p = 0.45$ and $b = 1.0$. Why does the push outcome affect the optimal bet size?

Exercise A.3 --- Fractional Kelly Sensitivity Analysis

A bettor estimates $p = 0.55$ for a bet at $-110$ (net odds $b = 0.909$). The full Kelly fraction is $f^* = (0.55 \times 0.909 - 0.45) / 0.909 \approx 0.055$.

(a) Calculate the theoretical growth rate $G(f)$ for Kelly fractions of 0.10, 0.20, 0.25, 0.33, 0.50, 0.75, and 1.00 (as fractions of full Kelly).

(b) Express each fractional Kelly growth rate as a percentage of the full Kelly growth rate. Which fraction captures 90% of the growth rate?

(c) Now suppose the bettor's probability estimate has a standard error of 0.03 (i.e., the true probability is $\hat{p} \sim N(0.55, 0.03^2)$). Simulate the expected growth rate for each Kelly fraction by sampling from this distribution. How does estimation uncertainty change the optimal fraction?

(d) Derive an analytical approximation for the optimal Kelly fraction under probability estimation uncertainty. (Hint: integrate the growth rate over the probability distribution.)

Exercise A.4 --- Kelly with Variable Bet Sizes

In practice, different bets offer different odds and different edges. A bettor has three simultaneous betting opportunities:

(a) Calculate the individual Kelly fraction for each bet.

(b) If the bets are independent, is it correct to bet the Kelly fraction on each simultaneously? What constraint should you impose on total allocation?

(c) Calculate the total bankroll percentage allocated if you bet full Kelly on all three. Is this practical?

(d) Propose a method for scaling down the Kelly fractions when the total allocation exceeds a maximum threshold (e.g., 20% of bankroll). Implement and test your method.

Exercise A.5 --- The Danger of Over-Kelly

Using a Monte Carlo simulation with $p = 0.55$ and odds $-110$:

(a) Simulate 10,000 paths of 500 bets each for bet sizes of $f = 0.06$ (full Kelly), $f = 0.09$ (1.5x Kelly), $f = 0.12$ (2x Kelly), and $f = 0.18$ (3x Kelly).

(b) For each bet size, calculate: median terminal wealth, mean terminal wealth, probability of ruin (defined as dropping below 10% of starting bankroll), and maximum drawdown distribution (mean, 95th percentile).

(c) Demonstrate that the median terminal wealth peaks at full Kelly and decreases for over-Kelly strategies, even though the mean terminal wealth continues to increase.

(d) Explain why the divergence between mean and median terminal wealth is the mathematical basis for fractional Kelly recommendations.

Exercise A.6 --- Bayesian Kelly Criterion

Suppose your probability estimate $\hat{p}$ is itself uncertain, modeled as $\hat{p} \sim \text{Beta}(\alpha, \beta)$.

(a) Write the expected growth rate $E[G(f)]$ by integrating the Kelly growth function over the Beta distribution.

(b) For a prior of Beta(55, 45) (representing a belief of roughly 55% with moderate confidence), find the optimal bet fraction numerically for a bet at $-110$ odds.

(c) Compare this "Bayesian Kelly" fraction to the standard Kelly fraction using the point estimate $p = \alpha / (\alpha + \beta)$.

(d) How does the Bayesian Kelly fraction change as you increase the uncertainty (e.g., Beta(11, 9) vs. Beta(55, 45) vs. Beta(550, 450))? Interpret the results.

Exercise A.7 --- Kelly for Correlated Sequential Bets

A bettor places a bet on the first half of an NBA game and is considering a correlated bet on the second half.

(a) If the first-half bet wins with probability 0.55 and the second-half bet's probability is 0.60 given a first-half win and 0.45 given a first-half loss, write the growth rate function for a strategy that bets fraction $f_1$ on the first half and fraction $f_2$ on the second half.

(b) Find the optimal $f_1$ and $f_2$ numerically (assume both bets pay at $-110$).

(c) Compare the growth rate of the correlated strategy to treating the bets as independent. What is the cost of ignoring the correlation?

(d) Under what conditions would ignoring correlation lead to over-betting?

Exercise A.8 --- Kelly Criterion for Parlays

A two-leg parlay has the following structure: Leg 1 wins with probability 0.55 (decimal odds 1.909), Leg 2 wins with probability 0.55 (decimal odds 1.909). The parlay pays $(1.909)^2 - 1 = 2.644$ net per unit wagered, and hits with probability $0.55 \times 0.55 = 0.3025$ (assuming independence).

(a) Calculate the Kelly fraction for this parlay using the standard formula.

(b) Compare the Kelly fraction and growth rate for: (i) the parlay, (ii) splitting the same bankroll across the two individual bets. Which strategy has the higher growth rate?

(c) Now suppose the two legs are positively correlated with $\rho = 0.20$, making the joint win probability 0.334 instead of 0.3025. Recalculate the Kelly fraction and growth rate for the parlay.

(d) At what correlation level does the parlay become superior to individual bets in terms of growth rate?

Exercise A.9 --- Kelly with Transaction Costs

In practice, the vig (vigorish) acts as a transaction cost. Standard $-110$ odds imply a 4.55% overround.

(a) Derive a modified Kelly formula that explicitly accounts for the vig. Express it in terms of your estimated true probability $p$ and the market's implied probability $p_m$ (including vig).

(b) For a bet at $-110$ where you estimate $p = 0.55$, calculate the Kelly fraction using both the standard formula and your modified formula. How do they compare?

(c) A bettor has access to two books: Book A offers $-110$ and Book B offers $-105$. Calculate the Kelly fraction at each book for the same $p = 0.55$. What is the value of having access to the better line?

(d) How much does the growth rate improve by moving from $-110$ to $-105$ odds? Express this as a percentage of the growth rate at $-110$.

Exercise A.10 --- Practical Kelly Implementation

Design a complete Kelly staking system for a bettor with a $10,000 bankroll who bets NFL spreads at $-110$.

(a) Define the inputs required for each bet (model probability, odds, current bankroll).

(b) Implement a function that calculates the bet size in dollars, applying quarter-Kelly with a maximum single-bet cap of 3% of current bankroll.

(c) The bettor places 10 bets per week for a 20-week NFL season. Simulate the season using true win probability of 0.54 and report: expected ending bankroll, probability of a losing season, worst expected drawdown, and 90% confidence interval for ending bankroll.

(d) Repeat the simulation with estimated probabilities that have a standard error of 0.02 (i.e., the bettor thinks their probability is 0.54 but it is sometimes higher and sometimes lower). How does this estimation error affect the results?

Part B: Portfolio Theory for Betting (10 exercises, 5 points each)

Exercise B.1 --- Expected Return and Variance of a Single Bet

For a bet with probability $p = 0.55$ and net odds $b = 0.909$ ($-110$):

(a) Calculate the expected return $\mu = pb - q$.

(b) Calculate the variance of the return $\sigma^2 = p(1-p)(b+1)^2$.

(c) Calculate the Sharpe ratio of this single bet (assume risk-free rate of 0).

(d) If you wager fraction $f$ of your bankroll, express the portfolio return and portfolio variance in terms of $f$, $\mu$, and $\sigma^2$.

Exercise B.2 --- Portfolio of Two Independent Bets

Consider two independent bets: Bet A has $p_A = 0.54$, $b_A = 0.909$, and Bet B has $p_B = 0.52$, $b_B = 1.50$.

(a) Calculate the expected return and variance for each bet individually.

(b) If you allocate fraction $f_A$ to Bet A and $f_B$ to Bet B, write the portfolio expected return and portfolio variance.

(c) Using mean-variance optimization with a risk aversion parameter $\lambda = 3$, find the optimal allocations $f_A^*$ and $f_B^*$ (subject to $f_A, f_B \geq 0$).

(d) Compare the portfolio Sharpe ratio to each individual bet's Sharpe ratio. Quantify the diversification benefit.

Exercise B.3 --- Covariance Between Correlated Bets

Two NFL bets are correlated because they involve teams in the same game: Bet A is "Home Team -3" and Bet B is "Over 45.5." Historically, the correlation between these outcomes is $\rho = 0.15$.

(a) If Bet A has $p_A = 0.54$ and $b_A = 0.909$, and Bet B has $p_B = 0.52$ and $b_B = 0.952$, calculate the covariance $\sigma_{AB} = \rho \cdot \sigma_A \cdot \sigma_B$.

(b) Write the portfolio variance including the covariance term.

(c) Find the optimal allocation using mean-variance optimization with $\lambda = 3$ and a total allocation constraint of 10%.

(d) Compare the result to the allocation assuming independence ($\rho = 0$). How much does the correlation reduce the diversification benefit?

Exercise B.4 --- The Efficient Frontier for Bet Portfolios

Given three available bets:

All pairwise correlations are 0.05.

(a) Construct the covariance matrix.

(b) Find five portfolios on the efficient frontier by varying the risk aversion parameter from 1 to 10.

(c) Plot (or describe) the efficient frontier in $(\sigma_P, \mu_P)$ space.

(d) Identify the portfolio that maximizes the Sharpe ratio. What are its allocations?

Exercise B.5 --- Diversification Across Sports

A bettor has edges in three sports with the following characteristics:

(a) Calculate the monthly expected return for each sport (edge times volume).

(b) Calculate the monthly return variance for each sport, accounting for within-sport correlation.

(c) Assuming cross-sport correlation of 0.01, calculate the variance of a combined portfolio that bets all three sports.

(d) Compare the Sharpe ratio of the combined portfolio to each individual sport. Which sport contributes the most to diversification?

Exercise B.6 --- Optimal Number of Simultaneous Bets

Using the diversification benefit formula from Section 14.2, analyze how many simultaneous bets optimize risk-adjusted returns.

(a) For identical independent bets with $p = 0.55$ and odds $-110$, compute the portfolio Sharpe ratio for $n = 1, 2, 5, 10, 15, 20, 30, 50$ simultaneous bets, assuming 10% total allocation.

(b) At what point do diminishing returns make additional bets not worth the effort? Define a threshold (e.g., less than 5% improvement in Sharpe ratio from adding one more bet).

(c) Now assume pairwise correlation of 0.05. How does this change the optimal number of bets?

(d) A practical bettor can thoroughly analyze 15 games per day. Is this sufficient to capture most of the diversification benefit? Justify with your calculations.

Exercise B.7 --- Same-Game Parlay Evaluation

Consider a same-game parlay with three legs from a single NFL game:

(a) Calculate the parlay odds assuming independence (multiply decimal odds).

(b) The correlation matrix is:

Using Monte Carlo simulation (Gaussian copula), estimate the true joint probability of all three legs hitting.

(c) Calculate the expected value of the parlay under both independence and the correlated assumption. Is the parlay positive EV?

(d) If the sportsbook applies a 20% margin to the parlay payout, recalculate the EV. At what correlation level would the parlay become negative EV?

Exercise B.8 --- Rebalancing Frequency

A bettor allocates their bankroll across three sportsbook accounts with target allocations of 35%, 35%, and 30%.

(a) After one month, the balances have drifted to 40%, 28%, and 32% due to wins and losses. Calculate the rebalancing trades needed.

(b) Each transfer between accounts takes 3 business days and costs $15 in fees. Calculate the cost of rebalancing as a percentage of total bankroll (assume $20,000).

(c) Define a threshold-based rebalancing policy: only rebalance when any account deviates by more than $X$% from target. Simulate one year of weekly balance changes (with 2% standard deviation per week) and calculate the optimal threshold that minimizes the total cost (rebalancing costs plus opportunity cost of sub-optimal allocation).

(d) How frequently should a $20,000 bettor rebalance versus a $200,000 bettor? Why does bankroll size matter?

Exercise B.9 --- Risk Parity Allocation

Risk parity allocates bankroll such that each bet contributes equally to total portfolio risk.

(a) For two bets with variances $\sigma_A^2 = 0.90$ and $\sigma_B^2 = 1.10$ and zero correlation, derive the risk parity allocation weights.

(b) Compare risk parity to equal allocation and to mean-variance optimal allocation (with $\lambda = 3$ and expected returns $\mu_A = 0.04$, $\mu_B = 0.03$).

(c) Under what conditions does risk parity outperform mean-variance optimization? When does it underperform?

(d) Apply risk parity to a portfolio of five bets with different variances. Does risk parity produce sensible allocations?

Exercise B.10 --- Portfolio Stress Testing

Design a stress test for a betting portfolio.

(a) Define three stress scenarios: (i) a 10-game losing streak, (ii) a systematic model error where true probability is 2% lower than estimated for all bets, and (iii) a market regime change where previously uncorrelated bets become correlated at $\rho = 0.20$.

(b) For a portfolio of 10 daily bets at quarter-Kelly sizing with a $20,000 bankroll, simulate each stress scenario over 30 days (10,000 paths each).

(c) Report the probability of a 20% drawdown and a 40% drawdown under each scenario.

(d) What bankroll sizing or portfolio adjustments would you recommend to survive the worst stress scenario with 95% confidence?

Part C: Drawdown Analysis and Recovery (10 exercises, 6 points each)

Exercise C.1 --- Maximum Drawdown Distribution

Write a simulation that estimates the distribution of maximum drawdown over 500 bets for a bettor with $p = 0.55$ and odds $-110$ at quarter-Kelly.

(a) Generate 20,000 bankroll paths and record the maximum drawdown of each path.

(b) Report the mean, median, 90th percentile, 95th percentile, and 99th percentile of the maximum drawdown distribution.

(c) What is the probability that the bettor experiences a drawdown exceeding 25%? Exceeding 40%?

(d) How do these numbers change if the bettor uses half-Kelly instead of quarter-Kelly?

Exercise C.2 --- Recovery Time Analysis

For a bettor at quarter-Kelly with $p = 0.55$ and odds $-110$:

(a) Calculate the expected number of bets to recover from drawdowns of 10%, 20%, 30%, and 50%.

(b) If the bettor places 10 bets per week, convert the recovery bets to calendar time.

(c) At 90% confidence, how many bets are needed to recover from a 20% drawdown?

(d) A bettor is 300 bets into the season and has experienced a 25% drawdown. Using your analysis, estimate the probability they will recover by the end of the season (200 more bets).

Exercise C.3 --- Drawdown-Adjusted Kelly

Design a dynamic staking strategy that reduces bet size during drawdowns.

(a) Define a drawdown-adjusted Kelly fraction: $f_{\text{adj}} = f^* \times \alpha \times (1 - \text{dd})^\gamma$ where $\text{dd}$ is the current drawdown, $\alpha$ is the base Kelly fraction, and $\gamma$ controls the aggressiveness of the reduction.

(b) Simulate this strategy with $\gamma = 0, 1, 2, 3$ over 1,000 bets. Compare the terminal wealth distribution and maximum drawdown distribution for each $\gamma$.

(c) Which value of $\gamma$ optimizes the ratio of median terminal wealth to 95th percentile maximum drawdown?

(d) Discuss the trade-offs of this approach. When might a drawdown-adjusted strategy underperform a fixed fractional strategy?

Exercise C.4 --- Comparing Staking Strategies

Implement and compare five staking strategies over 5,000 bets with $p = 0.54$ and odds $-110$:

(a) Flat 2% of initial bankroll.

(b) Flat 2% of current bankroll.

(c) Full Kelly.

(d) Quarter-Kelly.

(e) A hybrid: quarter-Kelly when above starting bankroll, flat 1% of current bankroll when in drawdown.

For each strategy, report: median terminal wealth, probability of doubling the bankroll, probability of ruin (below 10%), maximum drawdown (mean and 95th percentile), and Sharpe ratio of the equity curve.

Exercise C.5 --- The Gambler's Ruin Problem

A bettor starts with 100 units and bets 1 unit per bet at $-110$ with a true win probability of $p$.

(a) Calculate the probability of ruin (reaching 0 units before reaching 200 units) for $p = 0.50, 0.52, 0.54, 0.56$.

(b) How does the ruin probability change if the bettor starts with 200 units and the target is 400 units?

(c) Derive the expected number of bets until either ruin or target is reached for each $p$ value.

(d) Compare these results to a Kelly bettor (who can never technically reach 0 because bet sizes shrink proportionally). Why is the gambler's ruin framework still useful for practical bankroll planning?

Exercise C.6 --- Drawdown Policy Design

Design a three-tier drawdown policy for a professional bettor with a $50,000 bankroll.

(a) Specify the drawdown thresholds, the actions at each tier, and the conditions for returning to normal operations.

(b) Backtest your policy against 10,000 simulated seasons (500 bets per season, $p = 0.54$, odds $-110$, quarter-Kelly). How often does each tier trigger?

(c) Compare the terminal wealth distribution with and without the drawdown policy. Does the policy improve or reduce long-term returns?

(d) Modify your policy to account for the possibility that the bettor's edge has genuinely disappeared (i.e., $p$ has changed from 0.54 to 0.50). Design a statistical test the bettor should run at each drawdown tier to distinguish bad luck from lost edge.

Exercise C.7 --- Variance Drain

The concept of "variance drain" refers to the reduction in geometric growth rate caused by the volatility of returns, even when the arithmetic expected return is positive.

(a) For a bet with expected return $\mu$ and variance $\sigma^2$, show that the expected log return is approximately $\mu - \sigma^2 / 2$ for small bets.

(b) Calculate the variance drain for bets at odds $-110$ with $p = 0.55$ for bet sizes of 1%, 3%, 5%, and 10% of bankroll.

(c) At what bet size does the variance drain eliminate the growth rate entirely (i.e., expected log return equals zero)?

(d) How does this relate to the Kelly criterion? Show that Kelly is the bet size where the growth rate is maximized, which occurs before the variance drain overwhelms the edge.

Exercise C.8 --- Consecutive Loss Probability

A bettor at $p = 0.55$ is concerned about losing streaks.

(a) Calculate the probability of losing exactly $k$ bets in a row for $k = 5, 7, 10, 12, 15$.

(b) Over a 500-bet season, what is the expected number of losing streaks of length 5 or more? Length 8 or more?

(c) If the bettor uses quarter-Kelly starting at $\$10,000$, calculate the bankroll level after losing streaks of 5, 8, 10, and 12 consecutive bets.

(d) Develop a "streak risk" metric that combines streak probability with financial impact. Which streak length poses the greatest combined risk?

Exercise C.9 --- Drawdown Duration Analysis

The duration of a drawdown (time from peak to recovery) is often more psychologically challenging than the depth.

(a) Simulate 20,000 seasons of 500 bets ($p = 0.55$, $-110$, quarter-Kelly) and record both the depth and duration of the maximum drawdown in each season.

(b) Plot (or describe) the joint distribution of drawdown depth and duration. Are they correlated?

(c) What is the probability that the longest drawdown in a season exceeds 100 bets? 200 bets?

(d) A bettor who cannot psychologically tolerate a drawdown lasting more than 150 bets should use what Kelly fraction? Determine this through simulation.

Exercise C.10 --- Bankroll Sizing: How Much Do You Need?

For a bettor planning to bet NFL at quarter-Kelly with $p = 0.54$ and odds $-110$:

(a) Calculate the minimum bankroll needed to survive a 95th-percentile worst-case drawdown (from your Exercise C.1 results) while maintaining a minimum bet size of $50.

(b) If the bettor wants to earn $20,000 per year from betting, placing 15 bets per week over 20 weeks, what bankroll is required to achieve this expected profit at quarter-Kelly?

(c) Account for taxes: if the bettor's marginal tax rate on gambling income is 30%, recalculate the required bankroll.

(d) Present a complete "starting a professional betting career" financial plan, including initial bankroll, expected timeline to profitability, drawdown reserves, and minimum edge requirements.

Part D: Multi-Account Management (5 exercises, 6 points each)

Exercise D.1 --- Optimal Account Allocation

A bettor has accounts at five sportsbooks with the following characteristics:

Total bankroll: $25,000 with 15% reserve.

(a) Design a scoring function that ranks each book based on the four factors (juice, limits, restriction risk, promos).

(b) Calculate the optimal allocation to each book.

(c) One month later, Book D limits the bettor to $200 maximum bets. How should the allocation change?

(d) Book F launches with 3.5% juice, $8,000 limits, low restriction risk, and $400/month in promos. What is the new optimal allocation across all six books?

Exercise D.2 --- Line Shopping Value Calculation

A bettor has access to three books with different lines on the same game:

(a) If the bettor's model gives Team X a 55% chance of covering 3 points, calculate the EV at each book.

(b) Rank the books by EV and identify the best available bet.

(c) Over a 500-bet season, if the average line shopping improvement is 0.5 points of spread or 5 cents of juice, calculate the expected additional profit (assume $200 average bet size).

(d) Is it worth maintaining a smaller bankroll at a book with bad juice but high promotional value? Quantify the trade-off.

Exercise D.3 --- Seasonal Allocation Planning

Design a 12-month bankroll allocation plan for a bettor with a $30,000 bankroll and edges in NFL, NBA, MLB, and College Basketball.

(a) Specify the monthly allocation to each sport based on seasonal availability, expected edge, and expected bet volume.

(b) Identify the months with the highest and lowest total action. How should the reserve allocation change between peak and off-peak months?

(c) The bettor wants to increase their bankroll by 20% during the year. Calculate the monthly profit targets needed to achieve this goal.

(d) Present the plan as a monthly table showing: sports active, allocation per sport, expected bets, expected ROI, and cumulative profit projection.

Exercise D.4 --- Multi-Book Rebalancing Simulation

Simulate a 6-month period with five sportsbook accounts, starting with optimal allocations.

(a) Each week, simulate wins and losses at each book (with $p = 0.54$ and random allocation of bets to books based on line shopping). Track the balance at each book.

(b) Implement a threshold-based rebalancing policy (rebalance when any account deviates by more than 15% from target).

(c) Calculate the total number of rebalancing events over 6 months and the total transfer volume.

(d) Compare the final portfolio performance with rebalancing versus without rebalancing. Is the rebalancing effort justified?

Exercise D.5 --- Account Longevity Optimization

A bettor's accounts at recreational sportsbooks have an expected lifetime that depends on their win rate and bet patterns.

(a) Model account lifetime as an exponential distribution with rate parameter $\lambda = 0.005 \times \text{ROI\%}$ per bet (i.e., a bettor with 3% ROI has $\lambda = 0.015$ per bet). Calculate the expected number of bets before being limited for ROI of 1%, 2%, 3%, 5%.

(b) If the bettor varies their bet sizes randomly (between 50% and 150% of their target bet) and occasionally bets on recreational-looking plays (losing $\$50$ bets on popular favorites), how much longer can they extend account life? Model this quantitatively.

(c) Calculate the lifetime expected value of an account at a recreational book with 5% juice but generous limits, versus a sharp book with 3% juice but lower limits.

(d) Design an "account management" policy that specifies: bet size variation, bet frequency, prop betting frequency, and promotional participation to maximize expected lifetime account value.

Part E: Advanced Integration (5 exercises, 6 points each)

Exercise E.1 --- Complete Betting System Simulation

Build and simulate a complete betting system for one NFL season (18 weeks, ~15 bets per week).

(a) Generate model probabilities with a true edge of 2.5% (i.e., model probability is on average 2.5% above the break-even point).

(b) Apply quarter-Kelly staking with a maximum bet of 3% of current bankroll.

(c) Distribute bets across 4 sportsbook accounts using line shopping to achieve an average 0.3-point improvement.

(d) Report: season ROI, CLV, maximum drawdown, ending bankroll, number of rebalancing events, and total handle.

Exercise E.2 --- Risk-of-Ruin with Realistic Constraints

Extend the classical risk-of-ruin analysis to account for realistic constraints.

(a) A bettor has a $10,000 bankroll, bets at quarter-Kelly, and has a true $p = 0.54$ at $-110$. Calculate the theoretical risk of ruin using the standard formula.

(b) Now add: (i) a monthly withdrawal of $500 for living expenses, (ii) a tax payment of 25% on net profits every quarter, and (iii) a 10% chance of losing one sportsbook account per month (requiring redistribution). Simulate the risk of ruin under these realistic conditions.

(c) What minimum starting bankroll is needed to keep the 1-year ruin probability below 5% under these conditions?

(d) How does the answer change if the bettor increases their edge to 3.5%?

Exercise E.3 --- Kelly Meets Portfolio Theory

Derive the relationship between the Kelly criterion and mean-variance portfolio optimization.

(a) Show that for a single bet, the Kelly fraction $f^* = \mu / \sigma^2$ is equivalent to maximizing $\mu f - f^2 \sigma^2 / 2$ (the quadratic approximation of the log growth rate).

(b) For a portfolio of $n$ independent bets, show that the Kelly criterion applied to each bet independently is equivalent to the mean-variance optimal portfolio with risk aversion $\lambda = 1$.

(c) How does fractional Kelly (e.g., quarter-Kelly) correspond to a different risk aversion parameter? Derive the relationship.

(d) Discuss the practical implications: when is the Kelly framework more useful, and when is the portfolio framework more useful?

Exercise E.4 --- Monte Carlo Bankroll Projection Tool

Build a comprehensive Monte Carlo bankroll projection tool.

(a) The tool should accept: starting bankroll, number of bets per period, edge distribution (mean and standard deviation), odds distribution, Kelly fraction, and number of periods.

(b) Generate 10,000 bankroll paths and produce a "cone chart" showing the 10th, 25th, 50th, 75th, and 90th percentile bankroll trajectories.

(c) Calculate and report: expected ROI, probability of achieving various bankroll milestones (2x, 3x, 5x), expected time to reach each milestone, and risk of ruin at each time horizon.

(d) Validate the tool by comparing its output to the theoretical Kelly growth rate and variance formulas.

Exercise E.5 --- Annual Bankroll Management Plan (Challenge)

Create a complete annual bankroll management plan for a professional bettor with the following profile:

• Starting bankroll: $25,000
• Sports: NFL (3% edge), NBA (2.5% edge), MLB (3.5% edge), College BB (4% edge)
• Accounts at 4 sportsbooks
• Target annual profit: $15,000
• Risk tolerance: maximum 30% drawdown with 95% confidence

(a) Specify monthly allocations by sport and by book (12-month calendar).

(b) Set Kelly fractions for each sport that satisfy the drawdown constraint.

(c) Define a complete drawdown management policy with three tiers.

(d) Include rebalancing triggers, seasonal transitions, and contingency plans for account limitations.

(e) Simulate 10,000 seasons to validate that the plan achieves the profit target and satisfies the risk constraint.

Scoring Summary